The Bernoulli Brothers

The Bernoulli brothers were two outstanding mathematicians of the late 17th century and early 18th century. They were born in Basel, Switzerland and both graduated from Basel University. The elder brother, Jacob was offered a job as a professor at the university and Johann asked him to teach him mathematics. Their rivalry was born soon after and it is hard to tell whether or not it contributed to their success or not. They established an early correspondence with Gottfried Leibniz but weren’t just his disciples. They plowed their own way in the mathematical world, contributing to probability and infinitesimal calculus on their own. After Jacob died from tuberculosis in 1705 at the age of 51 in Basel, Johann took his position at the university. He went on to teach Leonhard Euler and Guillaume de l’Hopital. A well know problem that the brothers applied themselves to was the designing of a sloping ramp that would let a ball roll from top to bottom at the fastest possible speed. Johann demonstrated through calculus that neither a straight slope nor one with a very steep initial slope was the right answer. The answer was a curve with a less steep initial slope that was more optimal called a brachistochrone curve which is like an upside down cycloid. This is the perfect example of ‘the Calculus of Variations’ generalization that they developed together. This idea has been largely used in a number of fields such as engineering, financial investment, architecture and construction even space travel. Jacob Bernoulli’s book ‘the Art of Conjecture’ was published after his death, in 1714. It confirmed knowledge about probability theory and expected values as well as adding his personal ideas and thoughts. In the year 1685 he published two pamphlets, one on the parallels of logic and algebra and the other on probability. Another contribution of his was the term integral, which was used for the first time, with its integral meaning in a paper published in